This paper is about time preferences, the phenomenon that the very same things are usually considered the less valuable the farther in the future they are obtained. The utilities of those things are discounted at a certain rate. The paper presents a novel normative argument for exponential discount rates, whatever their empirical adequacy. It proposes to take indexical utility seriously, i.e. utilities referring to indexical propositions (that speak of ‘I’, ‘now’, etc.) as opposed to non-indexical propositions. Economic focus is (...) only on the latter, while the former are neglected. The potential ignorance of when is now generates a difference between indexical and non-indexical utility that can be exploited for a novel argument in favour of exponential discount rates. (shrink)

Under the presupposition that human time perception is distorted in intertemporal choice, this study constructs a time scale in the framework of axiomatic measurement. First, the conditions (homogeneity of degree one or two) to identify the form of a time scale are proposed so that one can determine whether the hyperbolic or exponential is a more suitable function for modeling people’s discounting. Homogeneity of degree one implies that subjective time delay is measured by a power scale and its (...) discount function becomes a power-exponential type, whereas homogeneity of degree two implies that subjective time delay is measured by a log scale and its discount function becomes a hyperbolic type. Second, a method is shown for constructing a model that can reflect subadditive and superadditive discounting. A non-commutative and non-associative concatenation operation is provided to generate a time string consisting of subdivided durations that incorporates the effect of repeated delay with a subdivided duration. The discounting model is yielded by substituting these strings in an exponential model equipped with a generalized time scale, and the determination of subadditivity or superadditivity depends on whether the discount function value of a product by the non-commutative and non-associative operation is, respectively, smaller or greater than that of a product by an operation of the usual extensive structure. (shrink)

According to Craig Callender [2022], the ‘received view’ across the social sciences is that, when it comes to time and preference, only exponential time discounting is rational. Callender argues that this view is false, even pernicious. Here I endorse what I take to be Callender’s main argument, but only in so far as the received view is understood in a particular way. I go on to propose a different way of understanding the received view that makes it true. (...) In short: When time discounting is suitably conceived, the exponential form of the discounting function is indeed uniquely rational. (shrink)

How should one discount utility across time? The conventional wisdom in social science is that one should use an exponential discount function. Such a function is a representation of the axioms that provide a well-defined utility function plus a condition known as stationarity. Yet stationarity doesnt really have much intuitive normative pull on its own. Here I try to cast it in a normative glow by deriving stationarity from two explicitly normative premises, both suggested by the philosophical thesis of (...) temporal neutralism. Putting the argument in this form helps us better understand exponential discounting and challenges to it. (shrink)

This paper challenges the conventional wisdom dominating the social sciences and philosophy regarding temporal discounting, the practice of discounting the value of future utility when making decisions. Although there are sharp disagreements about temporal discounting, a kind of standard model has arisen, one that begins with a normative standard about how we should make intertemporal comparisons of utility. This standard demands that in so far as one is rational one discounts utilities at future times with an (...) class='Hi'>exponential discount function. Tracing the justification for this standard through economics, philosophy and psychology, I’ll make what I believe is the best case one can for it, showing how a non-arbitrariness assumption and a dominance argument together imply that discounting ought to be exponential. Ultimately, however, I don’t find the case compelling. Non-exponential temporal discounting is often rational. If this is correct, Instead of trying to ‘fix’ non-exponential discounting because it is irrational and associated with negative life outcomes, we might instead focus attention on why the conditions obtain that make such discounting rational. (shrink)

Exponential discounted utility theory provides the normative standard for future discounting as it is employed throughout the social sciences. Tracing the justification for this standard through economics, philosophy and psychology, I’ll make what I believe is the best case one can for it, showing how a non-arbitrariness assumption and a dominance argument together imply that discounting ought to be exponential. Ultimately, however, I don’t find the case compelling, as I believe it is deeply flawed. Non-exponential (...) temporal discounting is often rational–indeed, the paragon of rationality. If this is correct, it’s an important point when considering policy interventions. Instead of trying to “fix” non-exponetial discounting because it is irrational and associated with negative life outcomes, we might instead focus attention on why the conditions obtain that make such discounting rational. (shrink)

Recent evidence on intertemporal choice suggests that decision-makers may exhibit both increasing and decreasing impatience simultaneously, called inverse-S discounting. This paper introduces trichotomic discounted utility to model inverse-S discounting. Under trichotomic discounted utility the decision-maker distinguishes between short-term delays, medium-term delays, and long-term delays. Exponential discounting holds within each category, but not necessarily across each category. We provide preference foundations for trichotomic discounted utility in the timed outcomes framework and in the consumption streams framework. The key (...) axiom is a weakening of the time consistency axiom that allows for both increasing and decreasing impatience. We also consider a weak, local version of time consistency, and show that this plausible condition is satisfied only by discount functions with exponential “pieces”. (shrink)

Time discounting is the phenomenon that a desired result in the future is perceived as less valuable than the same result now. Economic theories can take this psychological fact into account in several ways. In the economic literature the most widely used type of additive time discounting is exponential discounting. In exponential discounting, the fall of valuation depends by a constant factor on the length of the delay period. It is well known, however, that (...) exponential time discounting often does not describe well how people actually behave. Most people are averse to short delays in gratification now, while their future selves may not mind a bit of extra waiting. This behaviour can be described well by non-exponential discounting functions such as hyperbolic discounting. In hyperbolic discounting, valuations fall rapidly for small delay periods, but the fall gets slower for longer delay periods. Hyperbolic discounting captures phenomena such as procrastination, addiction and in general inconsistency over time. This chapter investigates whether forms of non-exponential discounting, in particular close to the so called Quasi-Hyperbolic model, could also be characterized in terms of dynamically consistent choices when individuals discount the welfare of future selves as well as their payoffs. (shrink)

I find much to like in Craig Callender's [2022] arguments for the rational permissibility of non-exponential time discounting when these arguments are viewed in a conditional form: viz., if one thinks that time discounting is rationally permissible, as the social scientist does, then one should think that non-exponential time discounting is too. However, time neutralists believe that time discounting is rationally impermissible, and thus they take zero time discounting to be the normative standard. (...) The time neutralist rejects time discounting because they think it is rationally impermissible to prefer to live a worse life in expectation because of arbitrariness. Callender’s attack on the time-neutralist position is the following: the time-neutralist’s non-arbitrariness intuition assumes the existence of nonexistent ‘pure’ time preferences. In response, I aim to clarify the time-neutralist position and show that the non-arbitrariness argument does not rely on the existence of pure time preferences. Instead, the debate between time neutralism and permissivism about time discounting boils down to a methodological question: can we ever criticize the content of preferences? If so, we should embrace time neutralism. (shrink)

While broadly in agreement with the conclusion that the exponentially discounted utility model (EDU) is not a universally valid rationality standard, I want to defend some intertemporal rationality criteria related to EDU, which Craig Callender might not share. My commentary explores the tension between these intuitions and Callender's arguments. In the first place, I show that many of the concerns that he raises are in fact compatible with intertemporal consistency (and sometimes even with EDU). Secondly, I rebut those arguments that (...) directly challenge consistency. I conclude that while we disagree on the rationality of consistency, we can nevertheless agree on its non-universality and its limited justification of policy interventions. (shrink)

Reinforcement schemes are a class of non-Markovian stochastic processes. Their non-Markovian nature allows them to model some kind of memory of the past. One subclass of such models are those in which the past is exponentially discounted or forgotten. Often, models in this subclass have the property of becoming trapped with probability 1 in some degenerate state. While previous work has concentrated on such limit results, we concentrate here on a contrary effect, namely that the time to become trapped may (...) increase exponentially in 1/x as the discount rate, 1− x, approaches 1. As a result, the time to become trapped may easily exceed the lifetime of the simulation or of the physical data being modeled. In such a case, the quasi-stationary behavior is more germane. We apply our results to a model of social network formation based on ternary (three-person) interactions with uniform positive reinforcement. (shrink)

Craig Callender [2022] provides a novel challenge to the non-arbitrariness principle. His challenge plays an important role in his argument for the rational permissibility of a non-exponential temporal discounting rate. But the challenge is also of wider interest: it raises significant questions about whether we ought to accept the non-arbitrariness principle as a constraint on rational preferences. In this paper, I present two reasons to resist Callender’s challenge. First, I present a reason to reject his claim that the (...) non-arbitrariness principle only targets pure time preferences. Second, I criticize the inference Callender draws from a modest claim to a much stronger claim. The modest claim is that it can be hard to reveal the contents of an agent’s preferences. The stronger claim is that this provides us with a reason to reject a certain kind of normative constraint on rational preferences. I argue that the modest claim doesn’t motivate the stronger claim. The upshot of my two arguments is good news for those sympathetic to the non-arbitrariness principle: Callender’s challenge can be overcome, at least as it currently stands. (shrink)

We conduct an experiment to investigate the degree to which deviations from exponential discounting can be accounted for by the hypothesis of hyperbolic discounting. Subjects are asked to choose between an earlier or later payoff in a series of 40 choice questions. Each question consists of a pair of monetary amounts determined by compounding a given base amount at a constant rate per period. Two bases (8 and 20 dollars), three compounding rates (low, medium and high) and (...) three delays (2, 4, and 6 weeks) are each used. There are also 2 initial periods (Today and 2 weeks) and there are two separate questionnaires, one with lower “realistic” compounding rates and the other with higher compounding rates, typical of those used in previous studies. We analyze the detailed patterns of choice in 6 groups of 6 related questions each (in which the base and rate is fixed but the initial period and delay varies), documenting the frequency of patterns consistent with exponential discounting and with hyperbolic discounting. We find that exponential discounting is the clear modal choice pattern in virtually all cases. Hyperbolic discounting is never the modal pattern (except in the sense that constant discounting is a special case of hyperbolic discounting). We also estimate a linear probability model that takes account of individual heterogeneity. The estimates show substantial increases in the probability of choosing the later option when the compounding rate increases, as one would expect. There are small, sometimes significant, increases in this probability when the delay is increased or the initial period is in the future. Such behavior is consistent with hyperbolic discounting, but can account for only a small proportion of choices. Overall, deviations from exponential discounting appear to be due to error, or to other effects not accounted for by hyperbolic discounting. Principal among these is an increase in later choices when the base is larger. (shrink)

Temporal discounting is the tendency to devalue temporally distant rewards. Past studies have examined the k-value, the indifference point, and the area under the curve as dependent measures on this task. The current study included these three measures and a fourth measure, called the interest rate total score. The interest rate total score was based on scoring only those items in which the delayed choice should be preferred given the expected return based on simple interest rates. In addition, associations (...) with several individual difference measures were examined including intelligence, executive functions (inhibition, working memory, and set-shifting), thinking dispositions (Need for Cognition and Consideration of Future Consequences) and engagement in substance use and gambling behavior. A staircase temporal discounting task was examined in a sample of 99 university students. Replicating previous studies, temporal discounting increased with longer delays to reward and decreased with higher reward magnitudes. A hyperbolic function accounted for variance in temporal discounting better than an exponential function. Reaction time at the indifference point was significantly longer than at the other choice points. The four dependent measures of temporal discounting were all significantly correlated and were also significantly associated with our individual difference measures. That is, the tendency to wait for a larger delayed reward on all of the temporal discounting measures was associated with higher intelligence, higher executive functions and more consideration of future consequences. Associations between our measures of temporal discounting and outcomes related to substance use and gambling behavior were modest in our university sample. (shrink)

One’s generosity to others declines as a function of social distance, which is known as social discounting. We examined cultural similarities and differences in social discounting and the mediating roles of the two aspects of interdependence (self-expression and distinctiveness of the self) as well as the two aspects of independence (harmony-seeking and rejection avoidance). Using the same procedure that previous researchers used to test North Americans, Study 1 showed that compared to North Americans, Japanese discount more steeply a (...) partner’s outcomes compared to their own future outcomes, whereas the decrease in the subjective value of the partner’s outcomes accelerates less as a function of social distance. To examine the cultural similarities and differences in social discounting in more detail, Study 2 tested Japanese and Germans and found that the hyperbolic with exponent model fitted the participants’ discounting behaviors better than the other models, except for the loss condition in Germans where the utility of the q-exponential model was indicated. Moreover, although the social discounting rate was higher in Japanese than in Germans, the cultural difference was limited to the gain frame. However, the decline in a person’s generosity accelerated less as a function of social distance in Japanese than in Germans. Furthermore, the cultural difference in the social discounting in gains was mediated by the level of harmony-seeking, which was higher in Germans than in Japanese. Implications for individuals’ generosity against the backdrop of cultural characteristics are discussed. (shrink)

We question the behavioral premise underlying Ainslie's claims about hyperbolic discounting theory. The alleged evidence for humans can be easily explained as an artefact of experimental procedures that do not control for the credibility of payment over different time horizons. In appropriately controlled and financially motivated settings, human behavior is consistent with conventional exponential preferences.

Nowadays, most loan transactions are contracted by using the exponential discounting as the underlying standard economic model to value this type of financial operations. In a framework of absence of fees to be paid by the borrower, the interest rate of the exponential discount function is, moreover, the true interest rate of the operation. Nevertheless, there exist a set of circumstances which make this identity false. Among others, these characteristics are: the use of linear discount as the (...) underlying discount function, splitting time when using a nominal interest rate, and the existence of fees in a loan at 0% interest rate. All these cases will be analyzed in this paper in the context of the so-called quick loans. (shrink)

This paper investigates two assumptions of the exponential discounted utility theory (EDU) to which Callender draws our attention: namely that we can cleanly distinguish pure from impure temporal preferences, and that past discounting can be ignored. Drawing on recent empirical work in this area, we argue that in so far as one might have thought that past-directed preferences are more pure than future ones, then there is evidence that people’s pure preferences (in so far as we can make (...) sense of that notion) show more interpersonal variation than has previously been thought. (shrink)

Craig Callender attempts to overturn conventional wisdom within decision theory by contending that rational intertemporal choices need not always conform to an exponential discounting function. He argues that there are cases in which hyperbolic discounting is the height of rationality. This paper does not seek to undermine Callender’s conclusions, but instead raises two interrelated theoretical concerns with his way securing them. The first concern is with his dismissal of influential dual-system explanations of rationality. It is argued that (...) Callender’s criticisms of said explanations fail to assess them at anything like their full weight. It is then suggested that dual-systems approaches in cognitive science do have substantial theoretical problems, but that Callender fails to identify them. The second concern builds on the first and suggests that Callender’s arguments against consensus normative standards might motivate a more dramatically reimagined notion of rationality than the one he seems to embrace. (shrink)

Intertemporal choice is the study of how people make choices about what and how much to do at various points in time, when choices at one time influence the possibilities available at other points in time. These choices are influenced by the relative value people assign to two or more payoffs at different points in time. Most choices require decision-makers to trade off costs and benefits at different points in time. These decisions may be about savings, work effort, education, nutrition, (...) exercise, health care and so forth. Since early in the twentieth century, economists have analyzed intertemporal decisions using the discounted utility model, which assumes that people evaluate the pleasures and pains resulting from a decision in much the same way that financial markets evaluate losses and gains, exponentially 'discounting' the value of outcomes according to how delayed they are in time. Discounted utility has been used to describe how people actually make intertemporal choices and it has been used as a tool for public policy. Policy decisions about how much to spend on research and development, health and education all depend on the discount rate used to analyze the decision. (shrink)

The problem of intertemporal choice arises when outcomes are received in different moments of time. This paper presents an axiomatic model of intertemporal choice when consumption in the previous moment of time contributes to utility evaluation of consumption in the current moment. This model generalizes classic discounted utility theory (also known as constant or exponential discounting) in two ways. First, in every moment of time, a decision maker derives utility not only from current consumption but also from “residual” (...) consumption in the previous moment of time. Second, these utilities are discounted with weights that are essentially a quasi-hyperbolic discounting function. The paper presents an application of the proposed model to the problem of optimal consumption and savings given a fixed income (wealth). When a decision maker derives satisfaction from both instantaneous consumption as well as a share of consumption in the previous moment of time, optimal consumption path is cyclic—periods of relatively high consumption are interchanged with periods of relatively low consumption. These cycles decay over time. Asymptotically, the consumption path exhibits conventional properties (constant, increasing or decreasing over time when a gross interest rate multiplied by discount factor is correspondingly equal to, greater than, or less than one). (shrink)

Behavioral science has long been puzzled by the experience of temptation, the resulting impulsiveness, and the variably successful control of this impulsiveness. In conventional theories, a governing faculty like the ego evaluates future choices consistently over time, discounting their value for delay exponentially, that is, by a constant rate; impulses arise when this ego is confronted by a conditioned appetite. Breakdown of Will presents evidence that contradicts this model. Both people and nonhuman animals spontaneously discount the value of expected (...) events in a curve where value is divided approximately by expected delay, a hyperbolic form that is more bowed than the rational, exponential curve. With hyperbolic discounting, options that pay off quickly will be temporarily preferred to richer but slower-paying alternatives, a phenomenon that, over periods from minutes to days, can account for impulsive behaviors, and over periods of fractional seconds can account for involuntary behaviors. Contradictory reward-getting processes can in effect bargain with each other, and stable preferences can be established by the perception of recurrent choices as test cases in recurrent intertemporal prisoner's dilemmas. The resulting motivational pattern resembles traditional descriptions of the will, as well as of compulsive phenomena that can now be seen as side-effects of will: over-concern with precedent, intractable but circumscribed failures of self-control, a motivated unconscious, and an inability to exploit emotional rewards. Hyperbolic curves also suggest a means of reducing classical conditioning to motivated choice, the last necessary step for modeling many involuntary processes like emotion and appetite as reward-seeking behaviors; such modeling, in turn, provides a rationale for empathic reward and the “construction” of reality. Key Words: altruism ; appetite; behavioral economics; classical conditioning; compulsions; dynamic inconsistency; emotions; empathy; freedom of will; hyperbolic discounting; impulsiveness; intertemporal bargaining; self-control; social construction; volition; weakness of will. (shrink)

We formulate and investigate experimentally a model of how individuals choose between time sequences of monetary outcomes. The model assumes that a decision maker uses, sequentially, two criteria to screen options. Each criterion only permits a decision between some pairs of options, while the other options are incomparable according to that criterion. When the first criterion is not decisive, the decision maker resorts to the second criterion to select an alternative. We find that: (1) traditional economic models based on (...) class='Hi'>discounting alone cannot explain a significant (almost 30%) proportion of the data no matter how much variability in the discount functions is allowed; (2) our model, despite considering only a specific (exponential) form of discounting, can explain the data much better solely thanks to the use of the secondary criterion; (3) our model explains certain specific patterns in the choices of the “irrational” people. We reject the hypothesis that anomalous behavior is due simply to random “mistakes” around the basic predictions of discounting theories: deviations are not random and there are clear systematic patterns of association between “irrational” choices. (shrink)

The case for longtermism depends on the vast potential scale of the future. But that same vastness also threatens to undermine the case for longtermism: If the universe as a whole, or the future in particular, contain infinite quantities of value and/or disvalue, then many of the theories of value that support longtermism (e.g., risk-neutral total utilitarianism) seem to imply that none of our available options are better than any other. If so, then even apparently vast effects on the far (...) future cannot in fact make the world morally better. On top of this, some strategies for avoiding this problem of “infinitarian paralysis” (e.g., exponential pure time discounting) yield views that are much less supportive of longtermism. In this chapter, we explore how the potential infinitude of the future affects the case for longtermism. We argue that (i) there are reasonable prospects for extending risk-neutral totalism and similar views to infinite contexts and (ii) many such extension strategies will still support the case for longtermism, since they imply that when we can only effect (or only predictably affect) a finite, bounded part of an infinite universe, we can ignore the unaffectable rest of the universe and reason as if the finite, affectable part were all there is. (shrink)

The energy domain is still dominated by equilibrium models that underestimate both the dangers and opportunities related to climate change. In reality, climate and energy systems contain tipping points, feedback loops, and exponential developments. This paper describes how to create realistic energy transition management models: quantitative models that can discover profitable pathways from fossil fuels to renewable energy. We review the literature regarding agent-based economics, disruptive innovation, and transition management and determine the following requirements. Actors must be detailed, heterogeneous, (...) interacting, learning, and strategizing. Technology should be represented as a detailed and heterogeneous portfolio that can develop in a bottom-up manner, using endogenous feedback loops. Assumptions about discount rates and the social cost of carbon should be configurable. The model should contain interactions between the global, national, local, and individual level. A review of modelling techniques shows that equilibrium models are unsuitable and that system dynamics and discrete event simulation are too limited. The agent-based approach is found to be uniquely suited for the complex adaptive sociotechnical systems that must be modelled. But the choice for agent-based models does not mean a rejection of other approaches because they can be accommodated within the agent-based framework. We conclude with practical guidelines. (shrink)

Temporal discounting is a technical operation in climate change economics. When discount rates are positive, economic evaluation treats future benefits as less important than equivalent present benefits. This chapter explains and critically evaluates four different reasons economists have given for tying discount rates to the interest rates we observe in real-world markets. I suggest that while philosophers have correctly criticized three of these reasons, their criticisms of the fourth miss the mark. This is because philosophers have not taken heed (...) of the distinct analytical framework in which the fourth reason arises. (shrink)

Discounting future costs and benefits is often defended on the ground that our descendants will be richer. Simply to treat the future as better off, however, is to commit an ecological fallacy. Even if our descendants are better off when we average across climate change scenarios, this cannot justify discounting costs and benefits in possible states of the world in which they are not. Giving due weight to catastrophe scenarios requires energetic action against climate change.

This paper defends the claim that mere temporal proximity always and without exception strengthens certain moral duties, including the duty to save – call this view Robust Temporalism. Although almost all other moral philosophers dismiss Robust Temporalism out of hand, I argue that it is prima facie intuitively plausible, and that it is analogous to a view about special obligations that many philosophers already accept. I also defend Robust Temporalism against several common objections, and I highlight its relevance to a (...) number of practical policy debates, including longtermism. My conclusion is that Robust Temporalism is a moral live option, that deserves to be taken much more seriously in the future. (shrink)

In response to cases that involve tiny probabilities of huge payoffs, some argue that we ought to discount small probabilities down to zero. However, this paper shows that doing so violates Independence and Continuity, and as a result of these violations, those who discount small probabilities can be exploited by money pumps. Various possible ways of avoiding exploitation will be discussed. This paper concludes that the money pump for Independence undermines the plausibility of discounting small probabilities. Much of the (...) discussion on Independence generalizes to other views that also violate Independence. (shrink)

Classical utilitarianism is very demanding and entails some counter-intuitive implications in moral dilemmas such as the trolley problem in deontological ethics and the repugnant conclusion in population ethics. This article presents how one specific modification of utilitarianism can avoid these counter-intuitive implications. In this modified utilitarian theory, called ‘discounted’ or ‘mild’ utilitarianism, people have a right to discount the utilities of others, under the condition that people whose utility is discounted cannot validly complain against such discounting. A complaint made (...) by a utility discounted person is not valid if either the existence of the discounting people in that option is in a specific sense necessary or the existence of the discounted person is in a sense not necessary. According to mild utilitarianism, we should choose the option that maximizes the total validly discounted or complaint-free discounted utility, i e. the sum of everyone’s utility minus the complaint-free discounts. As there are two conditions that make a complaint invalid, this right to discount can be translated into two versions: the right to bodily autonomy and the right to procreation autonomy. The former right relates to the mere means principle in deontological ethics, the latter right is useful in avoiding the repugnant conclusion problem in population ethics. The possibility of democratically imposing an upper bound on the permissible amount of discounting is discussed. (shrink)

When assessing the cost effectiveness of health care programmes, health economists typically presume that distant events should be given less weight than present events. This article examines the moral reasonableness of arguments advanced for positive discounting in cost-effectiveness analysis both from an intergenerational and an intrapersonal perspective and assesses if arguments are equally applicable to health and monetary outcomes. The article concludes that behavioral effects related to time preferences give little or no reason for why society at large should (...) favour the present over the future when making intergenerational choices regarding health. The strongest argument for discounting stems from the combined argument of diminishing marginal utility in the presence of growth. However, this hinges on the assumption of actual growth in the relevant good. Moreover, current modern democracy may be insufficiently sensitive to the concerns of future generations. The second part of the article categorises preference failures (which justify paternalistic responses) into two distinct groups, myopic and acratic. The existence of these types of preference failures makes elicited time preferences of little normative relevance when making decisions regarding the social discount rate, even in an intrapersonal context. As with intergenerational discounting, the combined arguments of growth and diminishing marginal utility offer the strongest arguments for discounting in the intrapersonal context. However, there is no prima facie reason to assume that this argument should apply equally to health and monetary values. To be sure, selecting an approach towards discounting health is a complex matter. However, the life-or-death implications of any approach require that the discussion not be downplayed to merely a technical matter for economists to settle. (shrink)

Nebel argues for the Repugnant Conclusion via the “Intrapersonal Repugnant Conclusion,” on which certainty of a mediocre life is better for individuals than a sufficiently small chance of an excellent life. In this article, I deny that acceptance of the Intrapersonal Repugnant Conclusion leads us to the Repugnant Conclusion. I point out that on many views which avoid the Repugnant Conclusion we should discount very small probabilities down to zero. If we do, then Nebel’s crucial premise of Ex Ante Pareto (...) fails, because discounting at the individual level can fail to match up with discounting at the population level. (shrink)

This article is a critical survey of the debate over the value of the social discount rate, with a particular focus on climate change. The ma- jority of the material surveyed is from the economics rather than from the philosophy literature, but the emphasis of the survey itself is on founda- tions in ethical and other normative theory rather than highly technical details. I begin by locating the standard approach to discounting within the overall landscape of ethical theory, and (...) explaining the assumptions and simplifications that are needed in order to arrive at the model that is standard in the discounting literature. The article then covers the general theory of the Ramsey equation and its relationship to observed interest rates, arguments for and against a positive rate of pure time preference, the consumption elasticity of utility, and the effect of various sorts of uncertainty on the discount rate. Finally, it turns specifically to the ap- plication of this debate to the case of climate change, focussing on the recent controversy over the low discount rate used in the Stern Review of the Economics of Climate Change. (shrink)

We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

discounting the future' is one on which philosophers and economists have divergent professional views. There is a lot of talking at cross-purposes across the disciplinary divide here; but there is a fair bit of confusion (I think) within disciplines as well. My aim here is essentially clarificatory. I draw several distinctions that I see as significant: • between inter-temporal and intergenerational questions • between price (discount rate) and quantity (inter-temporal and intergenerational allocations) as the ethically relevant magnitude, and • (...) between price change and preference change as the primary instrument of change. I show that discounting does not violate the principle of inter-temporal and intergenerational neutrality, but I also cast some doubt on whether making adequate allowance for future generations has really been the problem that economists and philosophers seem to have taken it to be. Key Words: discount rates • intergenerational justice • future generations • `feasibility' analysis. (shrink)

The desirable gambles framework offers the most comprehensive foundations for the theory of lower pre- visions, which in turn affords the most general ac- count of imprecise probabilities. Nevertheless, for all its generality, the theory of lower previsions rests on the notion of linear utility. This commitment to linearity is clearest in the coherence axioms for sets of desirable gambles. This paper considers two routes to relaxing this commitment. The first preserves the additive structure of the desirable gambles framework and (...) the machinery for coherent inference but detaches the interpretation of desirability from the multiplicative scale invariance axiom. The second strays from the additive combination axiom to accommodate repeated gambles that return rewards by a non-stationary processes that is not necessarily additive. Unlike the first approach, which is a conservative amendment to the desirable gambles framework, the second is a rad- ical departure. Yet, common to both is a method for describing rewards called discounted utility. (shrink)

We present theories of gravitation and electric potentials with exponential dependence on the reciprocal distance. In the context of this kind of electric potential we investigate the dynamics of a relativistic electron interacting with a proton.

We study the deviations from the exponential decay law, both in quantum field theory (QFT) and quantum mechanics (QM), for an unstable particle which can decay in (at least) two decay channels. After a review of general properties of non-exponential decay in QFT and QM, we evaluate in both cases the decay probability that the unstable particle decays in a given channel in the time interval between t and t+dt. An important quantity is the ratio of the probability (...) of decay into the first and the second channel: this ratio is constant in the Breit-Wigner limit (in which the decay law is exponential) and equals the quantity Γ1/Γ2, where Γ1 and Γ2 are the respective tree-level decay widths. However, in the full treatment (both for QFT and QM) it is an oscillating function around the mean value Γ1/Γ2 and the deviations from this mean value can be sizable. Technically, we study the decay properties in QFT in the context of a superrenormalizable Lagrangian with scalar particles and in QM in the context of Lee Hamiltonians, which deliver formally analogous expressions to the QFT case. (shrink)

This paper studies the interaction between savagean uncertainty and time preferences. We introduce a variation of the discounted subjective expected utility model, where time preferences are state dependent. Before uncertainty is resolved, the individual is unsure about the discount factor that will be used, even when evaluating certain payoffs. The model can account for the present bias and diminishing impatience, even if the future is discounted geometrically. The present bias disappears when the immediate payoff becomes uncertain. Although preferences are not (...) stationary, choices may be time consistent. (shrink)

The combinatorial parity principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bipartition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the parity principle requires exponential-size bounded-depth Frege proofs. Ajtai previously showed that the parity principle does not have polynomial-size bounded-depth Frege proofs even with the pigeonhole principle as (...) an axiom schema. His proof utilizes nonstandard model theory and is nonconstructive. We improve Ajtai's lower bound from barely superpolynomial to exponential and eliminate the nonstandard model theory. Our lower bound is also related to the inherent complexity of particular search classes . In particular, oracle separations between the complexity classes PPA and PPAD, and between PPA and PPP also follow from our techniques. (shrink)

Recent experiments suggest that search direction causally affects the discounted valuation of delayed payoffs. Comparisons between options can increase individuals’ patience toward future payoff options, while searching within options instead promotes impatient choices. We further test the robustness and specificity of this relationship using a novel choice task. Here individuals choose between pairs of delayed payoffs instead of single delayed outcomes. We observe a relationship between search styles and temporal discounting that are the opposite of those previously reported. Integrators (...) – those who tend to compare attributes within alternatives – discount and choose more slowly than comparators – those who are more likely to compare between alternatives. This finding supports and augments the view that individuals’ search strategy is predictive of subsequent discount rates. In particular, the direction of this relationship is further modifiable based on the spatial layout and varying information within an individual’s decision-making environment. (shrink)

This paper deals with the exponential stabilization problem for a class of nonlinear switched systems with mixed delays under asynchronous switching. The switching signal of the switched controller involves delay, which results in the asynchronous switching between the candidate controllers and subsystems. By constructing the parameter-dependent Lyapunov-Krasovskii functional and the average dwell time approach, some sufficient conditions in forms of linear matrix inequalities are presented to ensure the exponential stability of the switched nonlinear system under arbitrary switching signals. (...) In addition, through the special deformation of the matrix and Schur complement, the controllers with asynchronous switching are designed. Finally, a numerical example and a practical example of river pollution control are provided to show the validity and potential of the developed results. (shrink)

The “high school algebra” laws of exponentiation fail in the ordinal arithmetic of sets that generalizes the arithmetic of the von Neumann ordinals. The situation can be remedied by using an alternative arithmetic of sets, based on the Zermelo ordinals, where the high school laws hold. In fact the Zermelo arithmetic of sets is uniquely characterized by its satisfying the high school laws together with basic properties of addition and multiplication. We also show how in both arithmetics the behavior of (...) exponentiation depends on whether the empty set is an element of the base. (shrink)

We present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers.We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and (...) rotundity conditions, using techniques from tropical geometry.We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds.Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties.Abstract prepared by Francesco Paolo GallinaroE-mail: [email protected]: https://etheses.whiterose.ac.uk/31077/. (shrink)

This paper formulates and axiomatizes utility models for denumerable time streams that make no commitment in regard to discounting future outcomes. The models address decision under certainty and decision under risk. Independence assumptions in both contexts lead to additive or multiplicative utilities over time periods that allow unambiguous comparisons of the relative importance of different periods. The models accommodate all patterns of future valuation. This discount-neutral feature is attained by restricting preference comparisons to outcome streams or probability distributions on (...) outcome streams that differ in at most a finite number of periods. (shrink)

Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers and Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under integration. In this paper, we will present a natural refinement of their definition that allows for stability results to hold within the wider class of [Formula: see text]-minimal structures. One of the main technical improvements is that we remove the requirement of definable Skolem functions from the proofs. (...) As a result, we obtain stability in particular for all intermediate structures between the semi-algebraic and the sub-analytic languages. (shrink)